Finite Element Analysis For Elastic Multi-Structure Problems Based On A New Zienkiewicz-Type Element

Elastic multi-structures are usually assembled by some elastic substructures of the same or different dimensions (bodies, plates, rods, etc.) with proper junctions, which are widely occured in engineering applications. The finite element methods, widely used in computational mechancis, are effective in solving such kind of problems. In the past few decades, many researchers have been devoted to the study on the above problems. By means of the existing research results in the literature, this thesis is intended to develop a new finite element method, based on the new Zienkiewic-type nonconforming element (due to Wang, Shi, and Xu), to solve static elastic multi-structure problems and their vibration problems effectively, and then consider how to realize fast solution of the resulting discrete problems via domain decomposition techniques.At first, a new finite element method (named the P₁-P₃-NZT FEM) is introduced for solving general elastic multi-structure stationary problems, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized by conforming linear elements, transverse displacements on rods and plates are discretized respectively by Hermite elements of third order and new Zienkiewicz-type element, and the discrete generalized displacement fields in individual elastic members are coupled together by some feasible interface conditions. The optimal error estimate in the energy norm is established for the method, which is also validated by some numerical examples.Then a nonoverlapping domain decomposition method (DDM) is proposed to solve general elastic body-plate problems, discretized by the P₁-NZT finite element method, where displacements on body, longitudinal displacements on plate are discretized using P₁ conforming elements, and transverse displacements on plate is discretized by the new Zienkiewicz-type element, respectively. The main novelty is that its convergence rate is optimal (independent of the finite element mesh size), even for the shape-regular finite element triangulation. This enables us to combine the method with adaptive techniques in practical applications. Some numerical results are contained to illustrate the computational performance of the method.Meanwhile, we apply our finite element method to vibration analysis of elastic multi-strucutres. Concretely, the semi and fully discrete finite element methods are proposed for vibration analysis of elastic plate-plate structures. In the space directions, the longitudinal displacements on plates are discretized by conforming linear elements, and the corresponding transverse displacements are discretized by the new Zienkiewicz-type element, leading to a semi-discrete finite element method for the problem under consideration. Furthermore, the second derivative in time is discretized by the second order central difference, leading to a fully discrete scheme. Two approaches for choosing the initial functions and the error analysis in the energy norm is provided. Numerical results are presented to illustrate the computational performance.Some concise summary and propects are contained in the final part of the thesis.